3.1188 \(\int \frac {e^{4 \tanh ^{-1}(a x)}}{(c-a^2 c x^2)^2} \, dx\)

Optimal. Leaf size=18 \[ \frac {1}{3 a c^2 (1-a x)^3} \]

[Out]

1/3/a/c^2/(-a*x+1)^3

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Rubi [A]  time = 0.04, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6140, 32} \[ \frac {1}{3 a c^2 (1-a x)^3} \]

Antiderivative was successfully verified.

[In]

Int[E^(4*ArcTanh[a*x])/(c - a^2*c*x^2)^2,x]

[Out]

1/(3*a*c^2*(1 - a*x)^3)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 6140

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^p, Int[(1 - a*x)^(p - n/2)*
(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])

Rubi steps

\begin {align*} \int \frac {e^{4 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {1}{(1-a x)^4} \, dx}{c^2}\\ &=\frac {1}{3 a c^2 (1-a x)^3}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 17, normalized size = 0.94 \[ -\frac {1}{3 a c^2 (a x-1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(4*ArcTanh[a*x])/(c - a^2*c*x^2)^2,x]

[Out]

-1/3*1/(a*c^2*(-1 + a*x)^3)

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fricas [B]  time = 0.42, size = 41, normalized size = 2.28 \[ -\frac {1}{3 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2/(-a^2*c*x^2+c)^2,x, algorithm="fricas")

[Out]

-1/3/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2)

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giac [A]  time = 0.63, size = 15, normalized size = 0.83 \[ -\frac {1}{3 \, {\left (a x - 1\right )}^{3} a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2/(-a^2*c*x^2+c)^2,x, algorithm="giac")

[Out]

-1/3/((a*x - 1)^3*a*c^2)

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maple [A]  time = 0.02, size = 16, normalized size = 0.89 \[ -\frac {1}{3 c^{2} a \left (a x -1\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^4/(-a^2*x^2+1)^2/(-a^2*c*x^2+c)^2,x)

[Out]

-1/3/c^2/a/(a*x-1)^3

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maxima [B]  time = 0.32, size = 41, normalized size = 2.28 \[ -\frac {1}{3 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^4/(-a^2*x^2+1)^2/(-a^2*c*x^2+c)^2,x, algorithm="maxima")

[Out]

-1/3/(a^4*c^2*x^3 - 3*a^3*c^2*x^2 + 3*a^2*c^2*x - a*c^2)

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mupad [B]  time = 0.93, size = 40, normalized size = 2.22 \[ \frac {1}{-3\,a^4\,c^2\,x^3+9\,a^3\,c^2\,x^2-9\,a^2\,c^2\,x+3\,a\,c^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x + 1)^4/((c - a^2*c*x^2)^2*(a^2*x^2 - 1)^2),x)

[Out]

1/(3*a*c^2 - 9*a^2*c^2*x + 9*a^3*c^2*x^2 - 3*a^4*c^2*x^3)

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sympy [B]  time = 0.29, size = 42, normalized size = 2.33 \[ - \frac {1}{3 a^{4} c^{2} x^{3} - 9 a^{3} c^{2} x^{2} + 9 a^{2} c^{2} x - 3 a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)**4/(-a**2*x**2+1)**2/(-a**2*c*x**2+c)**2,x)

[Out]

-1/(3*a**4*c**2*x**3 - 9*a**3*c**2*x**2 + 9*a**2*c**2*x - 3*a*c**2)

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